Unbounded function with finite integral

Question

Can we have an unbounded positive uniformly continuous function from R to R with a finite improper integral from $-\infty$ to $\infty$ ?

I know that there's a continuous positive function that does this (a sequence of shrinking triangles) but it's not uniformly continuous.

Answer 1

If $f$ is bounded, then there is one:

Try the Schwarz function $f(x)=e^{-1/(x^{2}+1)}$, its derivative is $f'(x)=e^{-1/(x^{2}+1)}(2x/(x^{2}+1)^{2})$ which is bounded.

Answer 2

Hint: Suppose there exists such a function. Then there is a sequence $x_n$ with $|x_n|\to \infty$ such that $f(x_n)>1$ for all $n.$ Think about the $\delta$ that goes with $\epsilon = 1/2$ in the definition of uniform continuity.